Algebra - Exponential Functions

Section 6.1 : Exponential Functions

3. Sketch each of the following.

$f\left( x \right) = {6^x}$
$g\left( x \right) = {6^x} - 9$
$g\left( x \right) = {6^{x + 1}}$

Show All Solutions Hide All Solutions

a Show Solution

We can build up a quick table of values that we can plot for the graph of this function.

$x$	$f\left( x \right)$
-2	$f\left( { - 2} \right) = {6^{ - 2}} = \frac{1}{{{6^{\,2}}}} = \frac{1}{{36}}$
-1	$f\left( { - 1} \right) = {6^{ - 1}} = \frac{1}{6}$
0	$f\left( 0 \right) = {6^0} = 1$
1	$f\left( 1 \right) = {6^1} = 6$
2	$f\left( 2 \right) = {6^2} = 36$

Here is a quick sketch of the graph of the function.

$The graph of $f\left( x \right) = {6^x}$ on $ - 2 \le x \le 2$. The graph is always positive, is nearly flat along the x-axis for negative $x$ and increases very steeply to the right. The points that were computed in the table above are marked on the graph and include $\left( {0,1} \right)$, $\left( {1,6} \right)$ and $\left( {2,36} \right)$.$

b Show Solution

For this part all we need to do is recall the Transformations section from a couple of chapters ago. Using the “base” function of $f\left( x \right) = {6^x}$ the function for this part can be written as,

\[g\left( x \right) = {6^x} - 9 = f\left( x \right) - 9\]

Therefore, the graph for this part is just the graph of $f\left( x \right)$ shifted down by 9.

The graph of this function is shown below. The blue dashed line is the “base” function, $f\left( x \right)$, and the red solid line is the graph for this part, $g\left( x \right)$.

$The graph of $g\left( x \right) = {6^x} - 9$. The blue dashed line is the “base” function $f\left( x \right) = {6^x}$ and the solid red line is $g\left( x \right)$, which is exactly the same shape shifted down by 9. The solid graph flattens out along the horizontal line $y = - 9$ on the left instead of along the x-axis and crosses the x-axis just to the right of $x = 1$.$

c Show Solution

\[g\left( x \right) = {6^{x + 1}} = f\left( {x + 1} \right)\]

Therefore, the graph for this part is just the graph of $f\left( x \right)$ shifted left by 1.

The graph of this function is shown below. The blue dashed line is the “base” function, $f\left( x \right)$, and the red solid line is the graph for this part, $g\left( x \right)$.

$The graph of $g\left( x \right) = {6^{x + 1}}$. The blue dashed line is the “base” function $f\left( x \right) = {6^x}$ and the solid red line is $g\left( x \right)$, which is exactly the same shape shifted left by 1. Both graphs flatten out along the x-axis on the left and increase steeply on the right, with the solid graph always one unit to the left of the dashed one.$

\(x\)	\(f\left( x \right)\)
-2	\(f\left( { - 2} \right) = {6^{ - 2}} = \frac{1}{{{6^{\,2}}}} = \frac{1}{{36}}\)
-1	\(f\left( { - 1} \right) = {6^{ - 1}} = \frac{1}{6}\)
0	\(f\left( 0 \right) = {6^0} = 1\)
1	\(f\left( 1 \right) = {6^1} = 6\)
2	\(f\left( 2 \right) = {6^2} = 36\)